c
\(2F\cos {45^{ \circ}} + F' = \frac{{M{v^2}}}{R}\,\,\,\,\left( {From\,figure} \right)\)

\(where\,F = \frac{{G{M^2}}}{{{{\left( {\sqrt 2 R} \right)}^2}}}\,and\,F' = \frac{{G{M^2}}}{{4{R^2}}}\)

\( \Rightarrow \frac{{2 \times G{M^2}}}{{\sqrt 2 {{\left( {R\sqrt 2 } \right)}^2}}} + \frac{{G{M^2}}}{{4{R^2}}} = \frac{{M{v^2}}}{R}\)

\( \Rightarrow \frac{{G{M^2}}}{R}\left[ {\frac{1}{4} + \frac{1}{{\sqrt 2 }}} \right] = M{v^2}\)

\(\therefore \,v = \sqrt {\frac{{Gm}}{R}\left( {\frac{{\sqrt 2  + 4}}{{4\sqrt 2 }}} \right)}  = \frac{1}{2}\sqrt {\frac{{Gm}}{R}\left( {1 + 2\sqrt 2 } \right)} \)